What do people mean by "measurable definition?"

Acknowledging your assessment of self-inflating intern verbosity--of which your ignoring of same is part and parcel to transforming their minds into physician minds--the .pdf offering to which you refer seems exemplary of a reasoning fallacy of stating a premise so as to anticipate an outcome--similar in everyday parlance to begging the question. If a "definition" is what it is by virtue solely of being so defined, and since a measure (as defined) and a metric (as defined) are synonymous, then your conclusion, as by implication, is indeed not nonsensical but instead is precisely perceptive. A definition cannot be measured that is, of itself, already measured, and innately so.

It is indeed true that a definition itself cannot be measurable (or fail to be measurable, either). What those who use the phrase have in mind would probably be better expressed by 'definition in terms of something measurable'.

It is reasonable to infer that the person who wrote that was discussing "a clear definition that supports measurable metrics that would allow two like systems to be compared against each other", and decided that the title "A Measurable Definition of Resiliency Using 'Mission Risk' as a Metric" was already a wordy enough title without wording it as "A Definition That Supports Measurable Metrics That Would Allow Two Like Systems To Be Compared Against Each Other of Resiliency Using 'Mission Risk' as a Metric". It is within the realm of normal English usage to precede a noun with an adjective that does not, strictly speaking, refer to that noun, but to things connected to it, especially in a context such as a title where brevity is a concern, and there are further opportunities, such as an abstract, to use more precise language. See such terms as "metonymy".

In an online pdf titled "A Measurable Definition of Resiliency Using 'Mission Risk' as a Metric" I read this in the abstract, "Part of the problem...lacks a clear definition that supports measurable metrics that would allow two like systems to be compared against each other."

Using 'Mission Risk' as a Metric is the problem, "Mission Risk" is not a metric. A metric is a objective measure of something that can be repeated and determined and remains the same to all those who use it as a means of measure/a metric.

OED Metric. 4. A system or standard of measurement; a criterion or set of criteria stated in quantifiable terms.

Thus a metre or a foot or a decibel, or litre is the same anywhere on earth and regardless of who does the measuring.

'Mission Risk' is not a metric. To me, the Mission Risk might be "20", but to you, it might be 40. As time passes, it might be that the "Mission Risk" was actually "Zero" or "100".

The lesson is that you cannot use things "as a metric" if they are not constant and unable to be objectively measured.

This is why we do not use rubber bands to measure distances.